Stock Trading: An Optimal Selling Rule

Trading in stock markets consists of three major steps: select a stock, purchase a number of shares, and eventually sell them to make a profit. The timing to buy and sell is extremely crucial. A selling rule can be specified by two preselected levels: a target price and a stop-loss limit. This paper is concerned with an optimal selling rule based on the model characterized by a number of geometric Brownian motions coupled by a finite-state Markov chain. Such a policy can be obtained by solving a set of two-point boundary value differential equations. Moreover, the corresponding expected target period and probability of making money and that of losing money are derived. Analytic solutions are obtained in one- and two-dimensional cases. Finally, a numerical example is considered to demonstrate the effectiveness of our method.